Course detail

Functional Analysis II

FSI-SU2 Acad. year: 2020/2021 Winter semester

Review of topics presented in the course Functional Analysis I.
Theory of bounded linear operators. Compact sets and operators.
Inverse and pseudoinverse of bounded linear operators.
Bases primer: orthonormal bases, Riesz bases and frames.
Spectral theory of self-adjoint compact operators.

Language of instruction

Czech

Number of ECTS credits

3

Supervisor

prof. Aleksandre Lomtatidze, DrSc.

Department

Institute of Mathematics

Learning outcomes of the course unit

Knowledge of basic topics of functional analysis, of the theory of function spaces and linear operators. Problem solving skill mainly in Hilbert spaces, solution by means of abstract Fourier series and Fourier transform.

Prerequisites

Differential and integral calculus. Basics in linear algebra, Fourier analysis and functional analysis.

Planned learning activities and teaching methods

The course is taught through lectures explaining the basic principles and theory of the discipline. Exercises are focussed on practical topics presented in lectures.

Assesment methods and criteria linked to learning outcomes

Course-unit credit will be awarded on the basis of student's activity in tutorials focussed on solving tasks/problems announced by the teacher, and/or alternatively due to an idividual in-depth elaboration of selected topic(s).
The attendance in tutorials is compulsory. Examinations at a regular date are written or oral, the examinations at a resit or alternative date oral only. Examinations assess student's knowledge of the theoretical background an his/her ability to apply acquired skills independently and creatively.

Aims

The aim of the course is to make students familiar with main results of linear functional analysis and their application to solution of problems of mathematical modelling.

Specification of controlled education, way of implementation and compensation for absences

Absence has to be made up by self-study and possibly via assigned homework.

The study programmes with the given course

Programme M2A-P: Applied Sciences in Engineering, Master's
branch M-MAI: Mathematical Engineering, compulsory

Type of course unit

 

Lecture

26 hours, optionally

Teacher / Lecturer

prof. Aleksandre Lomtatidze, DrSc.

Syllabus

1. Review: topological, metric, normed linear and inner-product spaces, revision, direct product and factorspace
2. Review: dual spaces, continuous linear functionals, Hahn-Banach theorem, weak convergence
3. Review: Fourier series, Fourier transform and convolution
4. Bounded linear operators and associated main results
5. Adjoint and self-adjoint operatots incl. othogonal projection
6. Riesz Representation Theorem and Banach-Steinhaus Theorem
7. Unitary operators, compact sets and compact operators
8. Inverse of bounded linear operators in Banach and Hilbert spaces
9. Pseudoinverse of bounded linear operators in Hilbert spaces
10. Bases primer: orthonormal bases, Riesz bases and frames
11. Spectral theory of self-adjoint compact operators, Hilbert-Schmidt Theorem
12. Examples and applications primarily related to the field of Fourier analysis and signal processing
13. Reserve

Exercise

13 hours, compulsory

Teacher / Lecturer

prof. Aleksandre Lomtatidze, DrSc.

Syllabus

Refreshing the knowledge acquired in the course Functional analysis I and practising the topics presented at the lectures by individually solving selected examples and/or problems.